HOU Disk Model Framework

• HOU Disk Model is a self-consistent analytical framework that simulates disk–halo interactions using coupled gravitational potentials.
• It integrates iterative numerical techniques with explicit distribution functions to minimize parametric degeneracies in rotation curve analysis.
• Its versatile applications range from galactic dynamics to circumstellar disks, providing insights into dark matter halo flattening and accretion processes.

The HOU Disk Model refers to a family of analytical and semi-analytical frameworks developed for the study of astrophysical disks across a range of contexts, including galactic dynamics, protoplanetary and circumstellar disks, hydrodynamic stability, and accretion environments near compact objects. Originally motivated by the need for self-consistent modeling in galactic disk–halo interactions, the HOU Disk Model has since been adapted and applied in various subfields, each time prioritizing explicit solutions, dynamically consistent gravitational coupling, and reduction of parametric degeneracies.

1. Self-Consistent Disk–Halo Modeling Framework

Initially formulated to overcome limitations in traditional parametric disk–halo decompositions, the HOU Disk Model introduces a fully self-consistent method for embedding a zero-thickness, exponential stellar disk within a nonspherical, isothermal dark matter halo. The total gravitational potential is constructed as

$\Phi_T(R, z) = \Phi_{DM}(R, z) + \Phi_D(R, z)$

with the dark halo density following a Maxwellian isothermal profile,

$$\rho_{DM}(R, z) = \rho_{DM}^0 \exp[-\Phi_T(R,z)/\sigma^2]$$

and the stellar disk as

$\rho_D(R, z) = (M/L)\, \mu(R) \, \delta(z)$

where $(M/L)$ is the disk mass-to-light ratio, and $\mu(R)$ is the observed photometric brightness. By introducing dimensionless variables normalized to the disk exponential scale length $h$, the Poisson equation is recast for iterative solution: $$\left[ \frac{1}{\xi}\frac{\partial}{\partial \xi} \left( \xi\frac{\partial}{\partial \xi} \right) + \frac{\partial^2}{\partial \zeta^2} \right]\psi_{DM} = -\alpha \exp[ \psi_{DM}(\xi, \zeta) + \psi_D(\xi, \zeta)]$$

$$\left[ \frac{1}{\xi}\frac{\partial}{\partial \xi} \left( \xi\frac{\partial}{\partial \xi} \right) + \frac{\partial^2}{\partial \zeta^2} \right]\psi_D = -\frac{\beta}{2} \hat{\Sigma}(\xi) \delta(\zeta)$$

The two dimensionless parameters, $\alpha$ (normalized central halo density) and $\beta$ (normalized disk central surface density), play central roles in characterizing the models (Amorisco et al., 2010).

2. Halo Distribution Functions and Generalizations

The dynamical consistency of the HOU Disk Model is ensured through the use of explicit distribution functions for the dark halo. The principal choice is a Maxwellian (isothermal) function: $$f_{DM}(E) = \frac{\rho_{DM}^0}{(2\pi\sigma^2)^{3/2}} \exp(-E/\sigma^2)$$ This distribution reproduces the empirically flat, outer rotation curves observed in disk galaxies. The framework allows adaptation to other physically motivated functions, such as anisotropic or truncated forms relevant for exploring deviations in simulated halos (Amorisco et al., 2010). In protoplanetary and circumstellar contexts, the HOU model has been adapted to encode oscillatory or power-law prescribed density profiles (e.g., $\rho(R) \propto R^{-1.2}$) combined with constraints from midplane temperature and stability criteria (Christodoulou et al., 2019).

3. Disk–Halo Empirical Conspiracy and Constraints

One of the most significant outcomes is the elucidation of the so-called disk–halo conspiracy in rotation curve decomposition. Empirically, it is found that the radius where the rotation curve reaches two-thirds of its asymptotic value, $R_\Omega$ (with $V(R_\Omega) = (2/3) V_\infty$), is tightly correlated with the disk’s exponential scale length $h$, specifically: $\frac{R_\Omega}{h} = 1.07 \pm 0.03$ This indicates a much stronger and systematic fine-tuning between the luminous and dark matter distributions than previously assumed. This result supports a picture in which inner disk and halo properties are not independent but are mutually constrained by the need to reproduce flat, featureless rotation curves, which follows naturally from the self-consistent approach (Amorisco et al., 2010).

4. Rotation Curve Modeling and Degeneracy Removal

The HOU Disk Model produces smooth, flat rotation curves across intermediate radii ($\sim$2.2$h$ to $7h$), with the rotation velocity defined by: $$V_{mod}(R) = \left[ R \, \left( \frac{\partial \Phi_T}{\partial R}\right)_{z=0} \right]^{1/2}$$ To quantify flatness, a diagnostic

$$\Theta(\alpha, \beta) = \int_{2.2}^{7} \left| \frac{1}{\hat{V}_\infty} \frac{\partial V_{mod}}{\partial \xi} \right| d\xi$$

is computed, revealing that self-consistent models effectively maximize the region in $(\alpha, \beta)$ parameter space where $\Theta$ is minimized. The strict mutual coupling of halo and disk fields eliminates the classic "maximum disk" vs. "sub-maximal disk" ambiguity: the degeneracy in mass-to-light ratio and central halo density is dramatically reduced, and unique, physically plausible fits to observed curves become available—as demonstrated, for example, with NGC 3198 (Amorisco et al., 2010).

5. Geometric Flattening of Dark Halos

A direct consequence of the coupled field equations is that the dark matter halo becomes significantly flattened in regions dominated by the disk’s gravitational influence. The ellipticity, defined by

$\epsilon(\xi) = 1 - \frac{v(\xi)}{\xi}$

(where $v(\xi)$ is the intersection of an equipotential contour on the symmetry axis), illustrates pronounced oblateness near the mid-plane, particularly inside one disk scale length ($h$). This underscores that the dark matter halo cannot be treated as a spherically symmetric background at disk scales; instead, its morphology is directly influenced by baryonic structure. This geometric response is essential for accurate modeling of local dynamics and vertical stability (Amorisco et al., 2010).

6. Extensions and Applications

The basic analytic and iterative numerical framework for the HOU Disk Model has enabled several extensions:

• Vertical Structure: Multi-component disk plus halo models account for stars, molecular/atomic gas, and external halo gravity, yielding vertically pinched, steeper-than-sech$^2$ density profiles. These models predict moderate radial disk flaring and allow the HI scale height to constrain halo shape and density (Jog, 2 Jul 2025).
• Protoplanetary/Protostellar Disks: Isothermal Lane–Emden equation-based HOU models explain observed "dark gaps" (candidate planet-forming regions) as density oscillation maxima, supporting in situ planet formation scenarios and enabling quantitative disk stability analysis (Christodoulou et al., 2019).
• Thin Disk Dynamics: In dynamically active or turbulent systems, the HOU Disk Model can be embedded in a "2.5-dimensional" framework where the vertical structure (disk thickness $H$ and vertical velocity) is consistently evolved alongside surface densities, improving the treatment of vertical oscillations and gravitational softening (Westernacher-Schneider, 2023).
• Compact Object Accretion: Analytical HOU disk solutions for accretion flows near black holes feature explicit, self-consistent prescriptions for density, temperature, and magnetic field. Confinement to conical surfaces yields geometrically thinner disks near the horizon, resulting in sharper high-order images and sensitive polarization features in ray-traced shadow images (Yang et al., 24 Oct 2025).
• Spectral Analysis: In quantum lattice models, the HOU framework extends Chambers’s formula to kagome geometries, providing analytical tools for flat band and magnetic field-dependent spectrum calculations (Helffer et al., 2014).

7. Broader Significance and Future Directions

The HOU Disk Model unifies several principles in disk dynamics:

• Dynamical self-consistency across all mass components
• Strong empirical and theoretical constraints on otherwise degenerate parameter spaces
• Analytical tractability permitting explicit parameter studies and wide-ranging application, from galactic to circumstellar and compact-object scales
• Flexibility to accommodate additional physics, including realistic vertical structure, multi-component couplings, and diverse distribution functions.

Future research will refine HOU-type models through improved observational constraints (e.g., Gaia, IFU surveys, JWST), exploration of time-dependent multi-scale behavior, the inclusion of non-thermal support and feedback processes, and increased synergy with high-resolution simulations. The combination of analytic insight and numerical flexibility makes the HOU Disk Model a foundational tool across theoretical and observational astrophysics.